In the art of electroerosive wire-cutting defined above, a thin continuous wire or flamentary electrode is commonly employed which has a thickness as small as 0.005 to 0.5 mm. In addition, an extremely small gap must be formed between the workpiece and the traveling wire electrode. This condition unavoidably imposes a restriction on the desired smooth and sufficient passage of the water medium through the machining site. Thus, only a small fraction of the water medium supplied to the region of the workpiece juxtaposed with the traveling wire electrode is actually allowed to enter and flow through the machining gap at a limited flow rate. Difficulty therefore arises for machining chips and gases produced at discharge sites to be carried away smoothly. As a result, arcing and short-circuiting tend to develop between the workpiece and the wire electrode to disturb the progress of cutting and often causes breakage of the wire electrode and impairs the cutting stability.